Fraudulent figures: The numbers that say when things are fishy

Use the quirks of mathematics and you can spot basic mistakes as well as dubious dealings, including freaky voting patterns

Benford’s law

What do absent aliens, dodgy dictators and financial fraudsters have in common? Benford’s law can help hunt them down.

Benford’s law states that lists of numbers related to some natural or human activities will contain a particular distribution of digits. If you take a list of the areas of river basins, say, or the figures in a firm’s accounts, there will always be more numbers that start with 1 than any other. Numbers starting with 2 are the next most common, then 3 and so on. A number will start with a 9 only 4.6 per cent of the time.

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The pattern was originally suggested by astronomer Simon Newcomb in the 1800s in a riveting analysis of how people use books of logarithm tables. In the 1930s, the engineer Frank Benford rediscovered the result.

Why on earth should Benford’s law exist? Drill down to the root cause of most natural processes and they depend on random things, like the jostling of atoms. That produces bell-shaped curves, where most of the values are in the middle. But if several natural phenomena are at play — which is the case in a huge number of fields — then it turns out that Benford’s law is what holds.